**Zero-dimensional Commutative Rings**
There have been significant advances in the
theory of zero-dimensional commutative rings in
the last decade, but there remain topics within
the area where there are wide gaps in the theory.
Two of these topics in which I am interested are
the topic of realizability of a family of fields
as the family of residue fields of a
zero-dimensional ring, and the topic of embeddability
of a commutative ring in a zero-dimensional ring.
Related to the second topic there are a number of
questions concerning decomposition of ideals as
an infinite intersection of primary ideals that
are of independent interest.
**Rings of Integer-Valued Polynomials**
Again this this an area in which much work has
been done during the last twenty years. My
primary interest here is in the problem of
identifying and/or characterizing elements in a
ring of integer-valued polynomials that are
strong 2-generators---that is, which have the
property that they serve as one of two
generators of each finitely generated ideal to
which they belong. This question is of interest
when the base ring under consideration consists
of algebraic integers, and has not been completely
resolved even in the case of the classical ring
of integer-valued polynomials on the ring Z of
rational integers. |